Passives and Parasitics
You already know that each issue of Circuit Cellar has several circuit schematics complete with lots of resistors, capacitors, inductors and wiring. But those passive components don’t behave as expected under all circumstances. In this article, George takes a deep look at the way these components behave with respect to their operating frequency.
To draw schematic diagrams, we use common component symbols, not always realizing that passive components do not behave as expected under all circumstances. Their electrical characteristics change when the operating frequency increases. Therefore, when we design circuits, we can treat those components as ideal within a limited frequency range only.
No passive component is ideal. Passives such as resistors, capacitors, inductors and wiring combine the three fundamental electrical characteristics: resistance, capacitance and inductance. For this reason, when selecting parts, we must consider how their characteristics fit the intended design. We have to make sure their parasitic characteristics could be either considered negligible within the design’s operating frequency. And if they’re not, they have to be compensated in some way.
Our topic is the behavior of passive components with respect to their operating frequency only. At this time, we shall not be considering other design constraints such as the operating voltage, power dissipation, magnetic characteristics and so forth. In this article I have not considered transformers. Transformers are passive components too, but I discussed them in some detail in the Circuit Cellar issues 302, 303 and 304 (September, October and November 2015) .
Determining values of parasitic characteristics is not always easy, as few manufacturers publish them. You could measure them or, with luck, you might be able to find some data on the Internet. Some catalogs state the maximum recommended operating frequency.
Let us begin our investigation by taking a closer look at an ordinary resistor. Figure 1 shows its actual schematic. The total resistance is the sum of the resistive element R plus the resistance of both leads Rlead. Rlead is usually a negligible fraction of R and could be ignored. Both leads also introduce a serial parasitic inductance and a parallel parasitic capacitance to the circuit. The frequency response of the impedance of two typical through-the-hole resistors is plotted in Figure 2.
Prevailing parasitic values of these through-the-hole resistors are 8 nH for the lead inductance and 0.3 pF for the capacitance. Figure 2 shows that the resistors’ impedances remain essentially constant up to about 10 MHz. As the frequency increases, the larger, 50 kΩ resistor (red trace) begins to behave like an RC network with the corner frequency f=1/RC. Consequently, large value resistors are not suitable for high frequency circuits. Their frequency response can be somewhat improved by cutting the leads as short as possible or, better yet, by using SMD (surface-mount device) chip resistors.
Small value resistors, such as the 50 Ω one shown by the blue trace in Figure 2, become dominated by their parasitic inductance. This is especially true for wire-wound resistors, which are totally useless for applications at high frequencies. In radio frequency (RF) engineering, 50 Ω resistance happens to be considered a common yardstick. For the selection of other resistor values we can use the rule of thumb:
where f is the operating frequency and R is the resistance in ohms.
Just like the resistors, capacitors must also be viewed as a combination of a dominating capacitance with parasitic inductance and resistance. This is illustrated in Figure 3.
It is apparent that a simple capacitor is a combination of parasitic inductance of the leads LLEAD with lead resistance RLEAD in series connecting to the capacitance C. We must also consider two parasitic resistances RDC and RAC which shunt the capacitance and alter the component’s performance. Figure 4 is the frequency response plot of the impedance of a typical 0.1 μF capacitor.
Due to its parasitic inductance of about 25 × 1012 H (25 pH), the graph in Figure 4 shows, there is a series resonance around 100 MHz. At that frequency the capacitor’s impedance reaches minimum.
Because the dielectric material is not an ideal insulator, RDC represents the dielectric leakage, that is a DC current flowing through the dielectric material. In solid capacitors such as, for example the ceramic ones, RDC is very large, reaching hundreds of megohms. But the leakage may become a significant factor with electrolytic and super capacitors where it can be in the order of microamps. RAC stands for the dielectric frictional loss. The loss is due to the polarization of the charge when an AC current flowing through the capacitor changes its polarity. In applications with low RAC the current may be excessive, causing the capacitor to heat up.
This eventuality must be taken into account when a low RAC capacitor is employed in high power AC or RF applications. In component specifications those two resistances are often combined and referred to as Equivalent Series Resistance (ESR). This is an important characteristic for applications using electrolytic capacitors. Ceramic, polystyrene and other solid capacitors are generally too small for such applications and their ESRs too small to be of a serious concern. Generally, the lower the ESR, the better. By and large tantalum capacitors have lower ESR than aluminum electrolytic ones. This is something to keep in mind especially when working with switching power supplies.
The real inductor’s schematic shown in Figure 5 is analogous to the real capacitor. In their ideal form both capacitors and inductors are purely reactive components.
Once again, we have to account for the lead resistance Rlead and the parasitic capacitance. And importantly, we must not forget the effects of the PCB traces. Using a chip inductor does help, but it is not a complete solution because we can’t eliminate the resistance of the PCB traces. The parasitic capacitance C causes losses and the LC combination will cause parallel resonance at some frequency.
Figure 6 is an impedance plot of a 1 mH inductor exposed to a frequency sweep. Its parasitic capacitance of 0.1 pF causes parallel resonance at about 16 MHz. Inductors with a ferromagnetic core possess limited frequency response when compared with air inductors. RCORE in Figure 5 expresses the core losses. Those losses are due to the non-linear hysteresis and eddy currents. They increase with frequency and are difficult to calculate.
All inductors are limited by the magnitude of the current they can carry. This limitation is brought about by the resistance of the wire from which they are wound and the size of their leads. In addition, inductors with a ferromagnetic core can handle only a certain maximum current before their core becomes saturated, which completely changes their characteristics.
Wiring is similarly affected by hidden parasitic characteristics. Figure 7a shows a regular wiring diagram, while Figure 7b illustrates how the wiring behaves in the real world. If the wire run is shorter than about one eighth of the wavelength λ of the signal it carries and the wires are not running in parallel or not in close proximity to each other, capacitance CWIRE can be omitted. The load characteristics are not considered in this diagram, but they do interact with the wiring too.
If the conductors run in parallel, such as in lamp cords, coaxial cables, twin leads and so on. the parasitic capacitance CWIRE between them may have to be considered, depending on the frequency they carry. Should the lead lengths exceed λ/8 of the signal, the wiring will have to be regarded as a transmission line. This usually is not a concern for most household wiring, where the majority of signals are of low frequency.
Power distribution wiring in homes and buildings is unlikely to behave as a transmission line because the wavelength λ of the 60 Hz mains is approximately 5,000 km (3,107 miles) or 6,000 km (3,728 miles) for 50 Hz nets. One eighth of the wavelength is then 625 km (388 miles) for 60 Hz and 750 km (466 miles) for 50 Hz. However, the situation is different for external power transmission. If we were to consider 20 kHz to be the highest frequency of an audio signal feeding speakers, its wavelength λ will be about 15 km (9.3 miles) and taking one eighth of λ, the critical lengths for the speaker wires will be 1.9 km (1.2 miles).
Most computer interfaces act as transmission lines. With the typical interface cable length between 1 m and 2 m (around 3’ to 6’) the critical frequencies for the transmission line behavior would be about 37 MHz and 18.5 MHz respectively. That’s why longer computer cables are usually more expensive and harder to get. At higher frequencies printed circuit boards suffer from the same parasitic effects as any other type of wiring. However, this would be a topic of its own. Next month we’ll take a closer look at transmission lines.
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References  as marked in the article can be found there.
PUBLISHED IN CIRCUIT CELLAR MAGAZINE • DECEMBER 2018 #341 – Get a PDF of the issue